Abstract: A pair of algebras with a subalgebra of is said to have the (Principal) Congruence Extension Property (abbreviated as PCEP and CEP, respectively) if every (principal) congruence relation of can be extended to . A pair of algebras , is constructed having PCEP but not CEP, solving a problem of A. Day. A result of A. Day states that if is a subalgebra of and if for any subalgebra of containing , the pair has PCEP, then has CEP. A new proof of this theorem that avoids the use of the Axiom of Choice is also given.